The kind of bacteria that decomposes a polar bear is the same kind that decomposes any other animal. Proteobacteria and Firmicutes take turns eating the dead body until only the skeleton remains. Because polar bears inhabit very cold regions, the overall process of decomposition is slowed.
The kind of Proteobacteria that initiates decomposition is known as "Pseudomonas" and requires oxygen. After some time has passed, the anaerobic Firmicutes bacteria, "Peptoniphilus" and "Clostridium", replaces "Pseudomonas" and continues the decomposition process. The major difference between decomposition of a polar bear and that of other animals that do not inhabit cold regions is the rate at which it progresses.